where \(\mathbf {m}= [m_1, m_2, \cdots , m_N]^\mathrm {T}\)  is the vector of failure masks.
We assume that a failure sparsity property holds; that is, the number of faulty reflecting elements is much smaller than the total number of elements [1]}, [2]}, [3]}, and hence most entries in \(\mathbf {m}\)  are unity.In practice, the failure masks \(\mathbf {m}\)  may not be a near all-one vector due to general hardware imperfections (i.e., \(\mathbf {m}-\mathbf {1}_{N \times 1}\)  may not be strictly sparse).
In view of that, we focus on detecting faults and treat hardware imperfections as additive noises in the measurement or Gaussian perturbations on the measurement matrix.
As analyzed in [4]}, a simple and effective way to cope with this issue is to adjust the regularization parameters in the proposed diagnostic techniques to handle the increased noise level.
